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Logic Proofs. Monday 9/29/08. Take out HW – “Drawing Conclusions”; Pick up handouts! Quiz handed back Wednesday Logic Proofs - Law of Detachment HW – 1. Essay Due Thursday! 2. Law of Detachment. Definitions:.

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- Take out HW – “Drawing Conclusions”; Pick up handouts!
- Quiz handed back Wednesday
- Logic Proofs
- - Law of Detachment
- HW – 1. Essay Due Thursday!
- 2. Law of Detachment

Definitions:

Logically equivalent: when two statements always have the same truth value.

Premise: the statement that is given and excepted to be true.

Conclusion: the statement that has come from the premises.

Law of Detachment(a law of inference)

- The Law of Detachment states that when two given premises are true, one a conditional and the other the hypothesis of that conditional it then follows that the conclusion of the conditional is true.
- If you are given a conditional and the hypothesis the conclusion is true.
- p q and p therefore q

End for today

- Take out HW – “Law of Detachment”; Pick up handout!
- Place #1 – 4 answers (truth tables) on board! HW pass
- 2. Discuss Quiz
- 3. Logic Proofs
- - The Law of Contrapositive
- - Formal Proof
- - Modus Tollens
- HW – 1. Essay Due Thursday!
- 2. Law of Contrapositive / Modus Tollens

Law of Contrapositive

- The Law of Contrapositive states that when a conditional premise is true, it follows that the contrapositive of the premise is also true.
- If the conditional is true the contrapositive is also true.
- p q then ~q ~p

Formal Proof: given premises that are true use laws of reasoning to reach a conclusion.

Two Column Proof:

Column 1: Statements

Column 2: Reasons

Each statement must have a reason and they are number in sequence.

Example:

Given: If Joanna saves enough money, then she can buy a bike. Joanna can not buy a bike.

Prove: Joanna did not have enough money.

Let m = “ Joanna saves enough money”

Let b = “ Joanna can buy a bike”

Given: m b, ~b

Prove: ~m

m b

~b

Reason

Given

Given

3. Law of Contrapositive (step 1)

3. ~b ~m

4. ~m

4. Law of Detachment (step 2 & 3)

End for today

1. Rewrite the following as an equivalent DISJUNCTION!

a. 2x ≠ 4

a. (2x < 4) ∨ (2x > 4)

b. (b - 2 < 2) ∨ (b - 2 > 2)

b. b - 2 ≠ 2

- Write the negation of each in simplest terms!
- A. 10 + x < 5
- B. I never fail quizzes.
- C. Math is logical.
- D. It is not true that I am a good baseball player.

A. 10 + x ≥ 5

B. I sometimes fail quizzes

C. Math is not Logical.

D. I am a good baseball player.

- Write the following in symbolic form using the letters given. In each case the letter is the positive value
- of the statement!

a. The absolute value of x is equal to 2 if and

only if x =2 or x = -2.

(p: absolute value of x is 2; q: x = 2; r: x = -2)

b. If I’m late, then I’ll get into trouble, and if I’m

not late, then I won’t get into trouble.. (l,t)

Answers placed on board!

- Take out HW – “Law of Contrapositive”; Place essay in box!
- Complete Pre-game Warm-up! See Quiz Review
- 3. Logic Proofs
- - Modus Tollens
- - Chain Rule (Law of Syllogism)
- HW – 1. Law of Contrapositive / Modus Tollens #21-30
- 2. Complete Chain Rule Worksheet

Law of Modus Tollens

Given: If the tickets are sold out (t), then we’ll wait for the next show. (w)

We do not wait for the next show.

Prove: The tickets were not sold out.

Write the problem in symbolic notation!

Law of Modus Tollens

Given:t →w t → w

~w ~ w

Prove:~ t ~ t

or [(t → w) Λ ~ w] → ~ t

Set up a truth table to prove!

Prove [(t → w) Λ ~ w] → ~ t

Prove [(t → w) Λ ~ w] → ~ t

[(t → w) Λ ~ w] → ~ t is a Tautology therefore a valid argument!

Law of Modus Tollens

- The law of Modus Tollens states when 2 given premises are true, one a conditional and the other the negation of the conclusion of that conditional, it then follows that the negation of the hypothesis of the conditional is true.
- Given a conditional and the negation of the conclusion then the negation of the hypothesis is true. (combining Law of Contrapositive and Law of Detachment)
- p q, ~q therefore ~p

Invalid Arguments

All premises are true but they do not all lead to a valid argument.

Example: pq

q

no conclusion

pq

~p

no conclusion

End for Today

- Take out HW – “Law of Contrapositive, Chain Rule worksheet”
- Complete Quiz Retake! See Quiz Reviewhandout
- 3. Logic Proofs
- - Discuss Valid arguments
- - Chain Rule (Law of Syllogism); Applications of chain rule
- - Law of Disjunctive Inference
- HW – 1. Law of Disjunctive Inference worksheet
- 2. Application of Chain Rule #2-40 (evens)

1. Rewrite the following as an equivalent DISJUNCTION!

-x + 5 ≠ -3

- Write the negation of each in simplest terms!
- A. x ≤ -5
- B. Monkeys do not live in trees.
- C. Math is never logical.

- Write the following in symbolic form using the letters given. In each case the letter is the positive value
- of the statement!

If I don’t go to practice and train hard, then I will not be prepared for the game next week and our team will not win (p,h,g,w)

Chain Rule (Law of Syllogism)

[(p → q) Λ(q → r)] → (p → r)

Chain Rule (Law of Syllogism)

[(p → q) Λ(q → r)] → (p → r)

[(p → q) Λ(q → r)] → (p → r)

Valid Arguments

An argument is valid if the implication (P1 P2 P3 P4 …. Pn) C.

Λ

Λ

Λ

Λ

Λ

Premises

Premises

Is a valid argument?

[(p → q) Λ ~ q] → ~ p

Law of Modus Tollens

Chain RuleLaw of Syllogism

- Chain rule states that when 2 given premises are true conditional such that the consequent (conclusion) of the 1st is the antecedent (hypothesis) of the 2nd it follows that a conditional formed using the antecedent of the 1st and the consequent of the 2nd is true.
- You can combine conditionals if the conclusion of one is the hypothesis of another.
- p q and q r then p r

Chain RuleExample

p : You study

q : You pass

r : You get a surprise

P1:

p q

If you study, then you will pass.

P2:

q r

If you pass, then you will get a surprise.

Chain RuleExample

p : You study

q : You pass

r : You get a surprise

P1:

p q

If you study, then you will pass.

P2:

q r

If you pass, then you will get a surprise.

p r

C:

If you study, then you will get a surprise.

Law of Disjunction

- The previous laws involved conditionals this one does not.
- Law of Disjunctive Inference states when 2 given premises are true, on a disjunction and the other the negation of one of the disjuncts it then follows the other disjunct is true.
p ν q or p ν q

~q ~p

p

q

End for Today

- Take out HW – “Law of Disjuctive Inference, Chain Rule worksheet & Chain Rule Applications” – Place proofs on the white board!
- Another Proof - #10 Disjunctive Inference
- 3. Logic Proofs
- - Double Negation
- - DeMorgan’s Law
- 4. Begin HW; Pass back Quizzes
- HW – 1. pg 86-87 (evens)
- 2. Parent Signatures (3/2- by tomorrow!)

Givens:

Don is first or Nancy is Second.

If Nancy is second, then Chris is third.

If Chris is third, then Pattie is fourth.

Pattie is not fouth.

Prove: Pattie is not fourth.

Negations

- Contrapositive,
~p q ~q ~(~p)

~p q ~q p

- Modus Tollens
{(~p q) Λ ~q} ~(~p)

{(~p q) Λ ~q} p

Law of Double Negation

- Law of double negation states ~(~p) and p are logically equivalent.
- (you do not need to apply this law, continue as we have been)

DeMorgan’s Laws

-Discovered by English mathematician

-Tells us how to negate a conjunction and disjunction

Complete the following truth table

Since ~(m^s) and ~mV~s are logically equivalent for all cases, the negation of a conjunction is the disjunction.

DeMorgan’s Law states

- The negation of a conjunction of 2 statements is logically equivalent to the disjunction of the negation of each of the two statements.
- The negation of a disjunction of 2 statements is logically equivalent to the conjunction of the negation of each of the 2 statements.
- ~(p ^ q) (~p V ~q)
- ~(pV q ) (~p ^~q)

End for today

Mike likes to read and play basketball.

We can conclude-

Mike likes to read.

Mike likes to play basketball.

Laws of Simplification

- The law of simplification states that when a single conjunctive premise is true, it follows that each of the individual conjucts must be true.
- p ^ q therefore p is true
- p ^ q therefore q is true

Law of Conjunction:

The law of conjunction states that when 2 given premises are true, it follows that the conjunction of these is true.

p

q

p ^ q

AB is perpendicular to CD

AB is the bisector of CD

AB is the perpendicular bisector of CD.

Law of Disjunctive Addition

- Law of disjunctive addition states that when a single premise is true, it follows that any disjunction that has this premise as a disjunct is also true.
p

p V q

We solve an equation and see that x = 5.

We may also include that the statement

x = 5 or x > 5 is

True.

We may also include that the statement

x = 5 or Ian has 3 eyes is

True.

Given the statement r, which is a valid conclusion:

→

1. r k 2. r k 3. r k 4. r k

V

^

- Take out HW – “Law of Simplification,…” – Place proofs on the white board!
- 2. Logic Proofs – Practice!!!
- HW – Logic Proof Quiz Friday!

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